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Derivation of resistivity

Derivation of Resistivity: Boltzmann Transport Equation

In the quantum mechanical view of metals, resistivity arises when the equilibrium distribution of electrons is perturbed by an external electric field. By applying the Boltzmann Transport Equation and the Relaxation Time Approximation, we can mathematically derive the fundamental formula for resistivity, linking microscopic scattering to macroscopic resistance.

[INSERT DIAGRAM OF FERMI SURFACE SHIFT HERE]

1. The Electron Gas in Equilibrium

Inside a metal, conduction electrons behave like a gas. Their energy \( \varepsilon \) is related to their momentum \( \mathbf{p} \) by:

\[ \varepsilon = \frac{p^2}{2m} \]

They populate energy states according to the Fermi-Dirac distribution \( f_0 \):

\[ f_0(\varepsilon) = \frac{1}{e^{(\varepsilon-\mu)/k_B T} + 1} \]

In this state, for every electron moving left, one moves right. The net current is zero.

2. The Boltzmann Transport Equation (BTE)

When we apply an electric field \( \mathbf{E} \), the electrons accelerate, and the distribution function \( f \) shifts. The BTE describes the evolution of this distribution:

\[ \frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{r}} f + \mathbf{F} \cdot \nabla_{\mathbf{p}} f = \left( \frac{\partial f}{\partial t} \right)_{\text{coll}} \]

For a steady state (no time variation), uniform material (no spatial gradient), and force \( \mathbf{F} = -e\mathbf{E} \), this simplifies to:

\[ -e\mathbf{E} \cdot \nabla_{\mathbf{p}} f = \left( \frac{\partial f}{\partial t} \right)_{\text{coll}} \]

3. Relaxation Time Approximation

We assume that collisions (with impurities or phonons) try to restore the distribution back to equilibrium \( f_0 \) over a characteristic time \( \tau \) (relaxation time):

\[ \left( \frac{\partial f}{\partial t} \right)_{\text{coll}} = -\frac{f - f_0}{\tau} \]

Combining this with the BTE:

\[ -e\mathbf{E} \cdot \nabla_{\mathbf{p}} f = -\frac{f - f_0}{\tau} \]

Rearranging for the deviation \( \delta f = f - f_0 \), and approximating \( f \approx f_0 \) on the left side (linearization for small fields):

\[ \delta f = e\tau \mathbf{E} \cdot \nabla_{\mathbf{p}} f_0 \]

Using the chain rule \( \nabla_{\mathbf{p}} f_0 = \frac{\partial f_0}{\partial \varepsilon} \mathbf{v} \):

\[ \delta f = e\tau (\mathbf{E} \cdot \mathbf{v}) \frac{\partial f_0}{\partial \varepsilon} \]

4. Computing Current Density

The current density \( \mathbf{J} \) is the integral of velocity over the perturbed distribution:

\[ \mathbf{J} = -e \int \mathbf{v} \, \delta f \, \frac{d^3p}{(2\pi\hbar)^3} \]

Substituting our expression for \( \delta f \):

\[ \mathbf{J} = -e \int \mathbf{v} \left[ e\tau (\mathbf{E} \cdot \mathbf{v}) \frac{\partial f_0}{\partial \varepsilon} \right] \frac{d^3p}{(2\pi\hbar)^3} \]

For isotropic materials, averaging over all directions allows us to use \( \mathbf{v}(\mathbf{v}\cdot\mathbf{E}) = \frac{1}{3}v^2\mathbf{E} \):

\[ \mathbf{J} = -\frac{e^2 \tau}{3} \mathbf{E} \int v^2 \frac{\partial f_0}{\partial \varepsilon} \frac{d^3p}{(2\pi\hbar)^3} \]

At standard metal temperatures, \( \frac{\partial f_0}{\partial \varepsilon} \) acts like a delta function, selecting only electrons at the Fermi energy. Solving this integral yields the electron number density \( n \). This simplifies the equation to:

\[ \mathbf{J} = \left( \frac{ne^2 \tau}{m} \right) \mathbf{E} \]

5. The Resistivity Formula

Comparing the result with Ohm's Law in microscopic form, \( \mathbf{J} = \sigma \mathbf{E} \), we get the conductivity \( \sigma \):

\[ \sigma = \frac{ne^2 \tau}{m} \]

Resistivity \( \rho \) is the reciprocal of conductivity:

\[ \rho = \frac{m}{ne^2 \tau} \]

Parameter Definitions

  • \( \rho \): Resistivity (measure of opposition to flow).
  • \( m \): Effective mass of the electron (determines inertia).
  • \( n \): Electron number density (concentration of charge carriers).
  • \( e \): Elementary charge.
  • \( \tau \): Relaxation time (average time between collisions).

Physical Interpretation: Resistivity is directly proportional to the mass of the carriers (heavier particles are harder to move) and inversely proportional to the density of carriers and the time between collisions. More scattering (smaller \( \tau \)) leads to higher resistivity.